Find the values of `lambda` if the following of lines perpendicular :
`(1-x)/(3) = (7y-14)/(3lambda)=(z+1)/(2)` and `(7-7x)/(3lambda) = y/1 = (1-z)/(5)`

Answer

To find the values of `lambda` for which the given lines are perpendicular, we need to follow these steps:

### Step 1: Write the equations of the lines in standard form
The given lines are:
1. \((1-x)/(3) = (7y-14)/(3\lambda) = (z+1)/(2)\)
2. \((7-7x)/(3\lambda) = y/1 = (1-z)/(5)\)

We can express these in the standard form:
- For the first line, we can write:
\[ \frac{x - 1}{-3} = \frac{y - 2}{\frac{3\lambda}{7}} = \frac{z + 1}{2} \]
From this, we can identify the direction ratios \(L_1, M_1, N_1\) as:
\[ L_1 = -3, \quad M_1 = \frac{3\lambda}{7}, \quad N_1 = 2 \]

- For the second line, we can write:
\[ \frac{x - 1}{\frac{3\lambda}{7}} = \frac{y - 0}{1} = \frac{z - 1}{-5} \]
From this, we can identify the direction ratios \(L_2, M_2, N_2\) as:
\[ L_2 = \frac{3\lambda}{7}, \quad M_2 = 1, \quad N_2 = -5 \]

### Step 2: Use the condition for perpendicular lines
For two lines to be perpendicular, the following condition must hold:
\[ L_1 L_2 + M_1 M_2 + N_1 N_2 = 0 \]

Substituting the values we found:
\[ (-3) \left(\frac{3\lambda}{7}\right) + \left(\frac{3\lambda}{7}\right)(1) + (2)(-5) = 0 \]

### Step 3: Simplify the equation
Now, simplifying the equation:
\[ -\frac{9\lambda}{7} + \frac{3\lambda}{7} - 10 = 0 \]
Combining the terms:
\[ -\frac{9\lambda}{7} + \frac{3\lambda}{7} = -\frac{6\lambda}{7} \]
Thus, we have:
\[ -\frac{6\lambda}{7} - 10 = 0 \]

### Step 4: Solve for `lambda`
Rearranging gives:
\[ -\frac{6\lambda}{7} = 10 \]
Multiplying both sides by -1:
\[ \frac{6\lambda}{7} = -10 \]
Now, multiplying both sides by 7:
\[ 6\lambda = -70 \]
Finally, divide by 6:
\[ \lambda = -\frac{70}{6} = -\frac{35}{3} \]

### Conclusion
The value of `lambda` for which the lines are perpendicular is:
\[ \lambda = -\frac{35}{3} \]